Which has a larger area

A triangle with dimensions 29 cm by 29 cm by 40 cm, or a triangle with dimensions 29 cm by 29 cm by 42 cm

Pls explain solution

Thanks

Guest Feb 12, 2016

#3**+10 **

They have exactly the same area......believe it, or not.....!!!!!

Using Heron's formula to prove this, we have, for the 29,29,40 triangle

sqrt [49(20)(20)(9) ] = about 420 cm^2

And for the 29,29, 42 triangle, we have

sqrt [ (50(21)(21)(8) ] = 420 cm^2

Here is the explanation for the derivation of Heron's Formula along with a calculator to verify the above results :

http://www.mathsisfun.com/geometry/herons-formula.html

CPhill
Feb 12, 2016

#1**0 **

These are isosceles triangles. Area=1/2 X base X height

So, one with base 42 would have a larger area.

Guest Feb 12, 2016

#2**+5 **

.....but the HEIGHT changes with the bigger BASE resulting in BOTH triangles having the same area.

Guest Feb 12, 2016

#3**+10 **

Best Answer

They have exactly the same area......believe it, or not.....!!!!!

Using Heron's formula to prove this, we have, for the 29,29,40 triangle

sqrt [49(20)(20)(9) ] = about 420 cm^2

And for the 29,29, 42 triangle, we have

sqrt [ (50(21)(21)(8) ] = 420 cm^2

Here is the explanation for the derivation of Heron's Formula along with a calculator to verify the above results :

http://www.mathsisfun.com/geometry/herons-formula.html

CPhill
Feb 12, 2016

#4**+10 **

Here's the visual proof of this.....

Draw a circle centered at the origin with a radius of 29 with the specified triangles inscribed in the bottom half of this circle

Triangle ECD is isosceles with sides = 29, a base of 42 and a height, GC = 20

Triangle BCA is iscosceles with sides = 29, a base of 40 and a height, FC = 21

So....area of ECD = [42 * 20]/2 = 840/2 = 420 cm^2

And BCA has an area of [ 40 * 21] / 2 = 840/2 = 420 cm^2

Thus area of ECD = area of BCA ....!!!!

CPhill
Feb 12, 2016

#6**0 **

One more comment about this problem....we will always have integer solutions and equal triangles when the quantties involved form Pythagorean Triples.......specifically......

If the isoceles sides of the triangles = "c" in the formula a^2 + b^2 = c^2

And if we let the base of one triangle = two times the length of one of the legs in the triple.....then the height will be equal to the remaining number in the triple.

CPhill
Feb 14, 2016